Kabul, Afghanistan (CNN) -- Nearly a dozen insurgents and a U.S. contractor were killed when a group launched an early morning attack on Bagram Air Base in Afghanistan on Wednesday, officials said.

Nine service members were wounded and a building received minor damages during the attack, which included rockets, small arms and grenades. Four of the slain insurgents were "intended suicide bombers," the military said.

U.S. Army spokesman Lt. Col. Clarence Count Jr. said the insurgents failed "to breach the perimeter" and were "unable to detonate their suicide vests."

"The quick defensive reaction by the Bagram security forces likely saved a lot of lives," he said.

The Taliban claimed responsibility for the attack. Zabiullah Mujahid, a Taliban spokesman, told CNN that 20 armed men wearing suicide vests stormed the base around 4 a.m. (7:30 p.m. Tuesday ET), with four them detonating at the entrances to allow the other men to move in. Mujahid said that a "major firefight" took place inside the base.

Army Specialist Christina Dion, a spokeswoman who both lives and works on the base, said personnel were ordered to bunkers when the base came under attack and were allowed to return to their huts and tents about 90 minutes later.

"We're always prepared to deal with attacks on our base," Counts said. "The response this morning was immediate."

The fighting eventually tapered off about 7:30 a.m. local time, said NATO spokesman Lt. Col Todd Breasseale.

Another Bagram spokeswoman, Maj. Virginia McCabe, could not say if flights into and out of the air base had been stopped and said that authorities are investigating how the incident started.

"Established security procedures are in place," she said, adding that personnel with "mission-essential" responsibilities were continuing to work.

Bagram is a heavily fortified base, its perimeter guarded by high fences and thick, concrete walls. Dion said she didn't believe anyone could have made their way into the main portion of the base.

Taliban fighters have lobbed rockets into the base in previous attacks.